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Journal of Physics Condensed Matter

An asymptotically exact mean-field approach for Hubbard X-operators

Alex C. Hewson, Allen L. Wasserman

Published Jan 16, 1989• 3 citations
Publisher

Abstract

The authors consider a systematic projection method for terminating the Green function equations of motion for Hubbard X-operators. This gives a set of mean fields that depend on the operators chosen. When applied to lowest order in the N-fold-degenerate U to infinity Anderson model an expression for the one-electron Green function is obtained that is exact for N=1 and in the limit N to infinity . This approach avoids the necessity of introducing auxiliary fields, such as constraint fields or slave bosons, to obtain asymptotically exact mean fields in the large-N limit and has general applicability in similar strongly correlated problems.

DOI
10.1088/0953-8984/1/2/008

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The authors consider a systematic projection method for terminating the Green function equations of motion for Hubbard X-operators. This gives a set of mean fields that depend on the operators chosen. When applied to lowest order in the N-fold-degenerate U to infinity Anderson model an expression for the one-electron Green function is obtained that is exact for N=1 and in the limit N to infinity . This approach avoids the necessity of introducing auxiliary fields, such as constraint fields or slave bosons, to obtain asymptotically exact mean fields in the large-N limit and has general applicability in similar strongly correlated problems.
We propose an efficient method for nonperturbative calculation of Green's function in a correlated electron system. The key idea of the method is to project out irrelevant operators having zero norm in the ground state, which we refer to as effective projection theory. We apply the method to a mesoscopic Anderson model and show that for a given wavefunction ansatz, equations of motion can be closed only by relevant operators, allowing easy calculation of the zero-temperature Green's function. It turns out that the resulting Green's functions reproduce exact limits of both weak and strong interactions. The accuracy is also verified for small systems by comparison with exact diagonalization results, revealing that effective projection theory captures the essential correlated features in the entire regime of interactions.
An approximate analytical scheme of the dynamical mean-field theory (DMFT), that is used for electron systems with Hubbard correlations and is exact in the limit of the infinite dimensionality of a space, is developed.The effective single-site problem arising in the framework of this method is formulated in terms of the auxiliary Fermi-field.The irreducible Green's function technique with the projecting on the Hubbard basis of Fermi-operators is used for its solution.A system of DMFT equations is obtained in the approximation which is a generalization of the Hubbard-III approximation and combines it with a self-consistent renormalization of the local electron levels.It is shown that the proposed approach includes as simple specific cases a number of known approximations (Hubbard-III, AA, MAA, . . . ) based on the assumption of the single-site structure of the electron selfenergy.
The Anderson impurity model for finite values of the Coulomb repulsion U is studied using a slave-boson representation for the empty and doubly occupied f level. In order to avoid well-known problems with a naive mean-field theory for the boson fields, we use the coherent-state-path integral representation to first integrate out the double-occupancy slave bosons. The resulting effective action is linearized using two-time auxiliary fields. After integration over the fermionic degrees of freedom, one obtains an effective action suitable for a 1/Nf{\mathit{N}}_{\mathit{f}}Nf​ expansion. The same problem remains concerning the constraint as in the infinite U case. For T\ensuremath{\rightarrow}0 and Nf{\mathit{N}}_{\mathit{f}}Nf​\ensuremath{\rightarrow}\ensuremath{\infty}, exact results for the ground-state properties are recovered in the saddle-point approximation. Numerical solutions of the saddle-point equations show that even in the spin-degenerate case Nf{\mathit{N}}_{\mathit{f}}Nf​=2 the results are quite good.
Abstract We consider the large‐spin‐degeneracy (N) version of the U = ∞ one‐band Hubbard model and present a systematic 1/N expansion method which is based on Hubbard's X‐operator formulation and a Baym‐Kadanoff expansion. We compare this alternative X‐operator approach with the customary time‐continuum slave‐boson (SB) functional‐integral treatment, which has recently been criticized. In particular, we give a detailed derivation of the density Green's function within the new approach and compare the leading‐order results with those as obtained with the commonly used SB treatment, finding total agreement.
We introduce a new approach to strongly correlated systems, based on equations of motion for composite ("quasiparticle") operators. Conserving approximations to the appropriate Green's functions are generated through functional differentiation. For illustration, we apply this scheme to the infinite-UUU, single-band Hubbard model. The analogue of the Hartree-Fock approximation becomes exact in the limit of infinite spin degeneracy (NNN), while the random-phase approximation leads to the corresponding 1N\frac{1}{N}N1​ fluctuations.